ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 501382, 6 pageshttp://dx.doi.org/10.1155/2013/501382

Research Article

Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform

S. K. Upadhyay,¹ Anuj Kumar,² and Jitendra Kumar Dubey²

¹DST-CIMS, Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India²DST-CIMS, Banaras Hindu University, Varanasi 221005, India

Received 16 July 2013; Accepted 13 August 2013

Academic Editors: T. Ozawa, W. Yu, and C. Zhu

Copyright © 2013 S. K. Upadhyay et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An asymptotic series of general symbol of pseudo-differential operator is obtained by using the theory of fractional Fourier transform.

1. Introduction

Namias [1] introduced fractional Fourier transform which is a generalization of Fourier transform. Fractional Fourier transform is the most important tool, which is frequently used in signal processing and other branches of mathematical sciences and engineering. The fractional Fourier transform can be considered as a rotation by an angle in time-frequency plane and is also called rotational Fourier transform or angular Fourier transform. The fractional Fourier transform [2, 3], with angle of a function , is defined by where The corresponding inversion formula is given by where the kernel Zayed [3] and Bhosale and Chaudhary [4] studied fractional Fourier transform of distributions with compact support. Pathak and others [5] defined the pseudo-differential operator involving fractional Fourier transform on Schwartz space and studied many properties.

Our main aim in this paper is to generalize the results of Zaidman [6] and to find an asymptotic series of general symbol of pseudo-differential operator involving fractional Fourier transform.

Now we are giving some definitions and properties which are useful for our further investigations.

Linearity of fractional Fourier transform is given as where and are constants and and are two input functions.

Let denote the class of measurable functions defined on such that where .

From [5], generalized Sobolev space involving fractional Fourier transform is defined by and .

The convolution of two functions and is defined [5, 7] as provided that the integral exists.

Let be a class of all measurable complex-valued functions which are defined on . Then, we assume the following properties.(i) exists for all and is bounded to mesaurable function.(ii)We define , then where is complex-valued function defined on , which is measurable in and for all and satisfies the estimate: where   .

Let be a strictly decreasing sequence; that is, as and such that for all , Let be an infinite sequence of function defined on .

Then, we define a function where is a sequence of positive real numbers such that as .

From (12), it is clear that , for , , and .

The global estimate of the above defined function and of remainders of order is given as

Theorem 1. Let be a sequence of positive real numbers such that the following inequalities: are satisfied for . In particular the estimates are as follows:

Proof. The proof of the above theorem is obvious from [6, pages 233-234].

Theorem 2. Let be a sequence of positive real numbers such that the following estimates: are satisfied for . In particular the estimates are as follows: where , , , and .

Proof. The proof of above theorem is also obvious by using the same arguments from [6, pages 133–135].

2. Asymptotic Expansion of Pseudo-Differential Operator Associated with General Symbol

Definition 3. Let be a general symbol belonging to . Then pseudo-differential operator associated with symbol is defined by where is defined in (3) .

Definition 4. An infinitely differentiable complex-valued function is member of if and only if for every choice of and of non-negative integers, it satisfies

Lemma 5. A function satisfies (19) if and only if

Lemma 6 (Peetre). For any real number t and for all , the estimate is satisfied.

Theorem 7. Let ; then one has the following relation: where , .

Proof. By the definition of fractional Fourier transform (1), we have

Theorem 8. If is a symbol and is the associated operator, then, one has the following relation: where , .

Proof. We have By linearity of fractional Fourier transform (5) we get Now using (1) and (22), we get the required result.

Theorem 9. Let be a symbol and the associated operator; then one has the following relation: where , .

Proof. Firstly from (7) we haveNow the argument of (15) yields Thus we have for every , . Now we consider the function , where for and for and is defined as Therefore, Using (21) we get From Theorem 2 we get Here by (10) for all and since , therefore .Thus, we have and the inequality This implies that So that Using (36) we get Therefore, Now using (30) and (38) we get This implies that This implies the required result (27).

Theorem 10. One has the following estimates:

Proof. Let , then by using (11) and (12) we find that for . Therefore, we have Now we define the operator given on by Hence, Thus, Since , is a linear operator into itself.

Definition 11. A linear operator with , and , there exists a constant such that The infimum of all orders of is called true order of .

Definition 12. Let be a linear operator from into itself and satisfy the following inequality: Then is said to be a canonical operator of degree , where .

Definition 13. Let be a strictly decreasing sequence of real numbers and a sequence of canonical operators of degree . Then, corresponding a sequence of positive real numbers and , a linear operator is asymptotically expanded into the series if it satisfies the following inequality:

Theorem 14. Let be a sequence of symbols belonging to and a strictly decreasing sequence of real numbers that tends to . Then, there exists a sequence of canonical operators of degree and a linear operator in such that (i)t.o ;(ii), that is, t.o  .

Proof. Note that (i) is obviously true by using Theorem 9 and for (ii) by using the arguments of [6, pages 241-242]. We can define the canonical operator by the following way: where belongs to class of symbol . Also we have Therefore, Using the previous definitions (12) and (13), we have From (15) and (16), we get , where and .Also . Now using these arguments in (52) we get Using Theorem 9, we get Hence, we get

Acknowledgments

The first author is thankful to DST-CIMS, Banaras Hindu University, Varanasi, India, for providing the research facilities, and the second author is also thankful to DST-CIMS, Banaras Hindu University, Varanasi, India, for awarding the Junior Research Fellowship from December 2012.

References

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” Journal of the Institute of Mathematics and its Applications, vol. 25, no. 3, pp. 241–265, 1980.

L. M. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Signal Processing Letters, vol. 42, no. 11, pp. 3084–3091, 1994.

A. I. Zayed, “Fractional Fourier transform of generalized functions,” Integral Transforms and Special Functions, vol. 7, no. 3-4, pp. 299–312, 1998.

B. N. Bhosale and M. S. Chaudhary, “Fractional Fourier transform of distributions of compact support,” Bulletin of the Calcutta Mathematical Society, vol. 94, no. 5, pp. 349–358, 2002.

R. S. Pathak, A. Prasad, and M. Kumar, “Fractional Fourier transform of tempered distributions and generalized pseudo-differential operator,” Journal of Pseudo-Differential Operators and Applications, vol. 3, no. 2, pp. 239–254, 2012.

S. Zaidman, “On asymptotic series of symbols and of general pseudodifferential operators,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 63, pp. 231–246, 1980.

R. S. Pathak, A Course in Distribution Theory and Applications, Narosa Publishing House, New Delhi, India, 2009.